# Why does the right hand rule works?

pconstantino
I am having a hard time understanding how come the right hand rule will tell you the direction of a vector that is perpendicular to other two vectors.

I mean, i've seen some examples of a current flowing through a wire, which created a magnetic field around it, if there is a positive particle traveling around it with a velocity vector "v" how is it possible that the right hand rule tells you the direction of the field?

Is it because many experiements agree with it?

This is so confusing.

Mentor
2021 Award
I am having a hard time understanding how come the right hand rule will tell you the direction of a vector that is perpendicular to other two vectors.

I mean, i've seen some examples of a current flowing through a wire, which created a magnetic field around it, if there is a positive particle traveling around it with a velocity vector "v" how is it possible that the right hand rule tells you the direction of the field?

Is it because many experiements agree with it?

This is so confusing.
The right hand rule works because we have all agreed on it. If we had all agreed on the left hand rule then the left hand rule would also work. This is similar to calling the charge on a proton "positive" and the charge on an electron "negative". It works because we all agree on it, but nothing would be different if we had adopted the opposite convention as long as we all agreed on that.

Naty1
Yes to Dalespams post...It's akin to asking "Why does ROYGBIV" give the order of lightd colors....

It's not like somebody dreamed up the acronym first, then discovered colors of light...we all agreed on the name of different colors,for consistency, for different frequencies, and then to help remember their order somebody came up with the "rule"....just a convenient convention.

pconstantino
but how come the electric field points in the correct direction?? the field doesnt know anything about our hands.

Mentor
2021 Award
I think you mean the magnetic field, not the electric field.

How do we know which direction the magnetic field points? We can determine that by looking at a moving charge and seeing which direction the force is.

USING RIGHT HAND RULE:
So let's say we have a vertical wire in front of us with current running up. By the right-hand rule that will cause a magnetic field circulating ccw around the wire. Now, consider a positive charge located to the right of the wire (field pointed away from us) which is moving upwards. By the right-hand rule the force on this charge is pointed towards the wire.

USING LEFT HAND RULE:
Again, let's use the same vertical wire in front of us with current running up. By the left-hand rule that will cause a magnetic field circulating cw around the wire. Now, again consider a positive charge located to the right of the wire (field pointed towards us) which is moving upwards. By the left-hand rule the force on this charge is again pointed towards the wire.

The convention doesn't matter. All that matters is that we all pick a convention and agree to stick with it.

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Staff Emeritus
Gold Member
but how come the [STRIKE]electric[/STRIKE] magnetic field points in the correct direction?? the field doesnt know anything about our hands.

No it doesn't, but the right hand rule works because the magnetic field is related to the velocity and position vectors by a vector cross product. This is expressed in the Biot-Savart Law for static magnetic fields. In the specific case of a single point charge moving at a constant velocity, the Biot-Savart law says that B is proportional to qvr where v is the velocity vector and r is a position vector going from the charge to the point in space where the magnetic field is being evaluated. So, the right hand rule just describes/comes from the vector cross product. The more fundamental question (and I think this is what you are getting at) is, "why is the Biot-Savart law true for magnetic fields?", or, expressed in other words, "why does the relationship between the magnetic field generated by moving charges and the velocity of those moving charges have this particular mathematical form?" I think the answer to those questions is, "because that is the mathematical relationship that has been shown to be true by experiment (or at least describes *well* all experimental observations to date/is consistent with them)."

EDIT: I added a "q" to my equation to make it clearer that the sign of the charge comes into play as well, so that if you redefine what is positive vs. negative charge, you change whether it's a right-hand or left-hand rule, as others have been telling you.

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pconstantino
No it doesn't, but the right hand rule works because the magnetic field is related to the velocity and position vectors by a vector cross product. This is expressed in the Biot-Savart Law for static magnetic fields. In the specific case of a single point charge moving at a constant velocity, the Biot-Savart law says that B is proportional to qvr where v is the velocity vector and r is a position vector going from the charge to the point in space where the magnetic field is being evaluated. So, the right hand rule just describes/comes from the vector cross product. The more fundamental question (and I think this is what you are getting at) is, "why is the Biot-Savart law true for magnetic fields?", or, expressed in other words, "why does the relationship between the magnetic field generated by moving charges and the velocity of those moving charges have this particular mathematical form?" I think the answer to those questions is, "because that is the mathematical relationship that has been shown to be true by experiment (or at least describes *well* all experimental observations to date/is consistent with them)."

EDIT: I added a "q" to my equation to make it clearer that the sign of the charge comes into play as well, so that if you redefine what is positive vs. negative charge, you change whether it's a right-hand or left-hand rule, as others have been telling you.

so do you mean that it all comes from the mathematical definition of cross product?

for example, when we take the cross product of 2 vectors via cross product and use determinants, etc... does the right hand rule always match this vector given in the mathematical operation?

if so that seems magical

Mentor
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if so that seems magical
Did you read my example? Nothing magical at all.

pconstantino
I think you mean the magnetic field, not the electric field.

How do we know which direction the magnetic field points? We can determine that by looking at a moving charge and seeing which direction the force is.

USING RIGHT HAND RULE:
So let's say we have a vertical wire in front of us with current running up. By the right-hand rule that will cause a magnetic field circulating ccw around the wire. Now, consider a positive charge located to the right of the wire (field pointed away from us) which is moving upwards. By the right-hand rule the force on this charge is pointed towards the wire.

USING LEFT HAND RULE:
Again, let's use the same vertical wire in front of us with current running up. By the left-hand rule that will cause a magnetic field circulating cw around the wire. Now, again consider a positive charge located to the right of the wire (field pointed towards us) which is moving upwards. By the left-hand rule the force on this charge is again pointed towards the wire.

The convention doesn't matter. All that matters is that we all pick a convention and agree to stick with it.

Sorry hadn't seen your reply. your are saying "by the right hand rule..." but the direction of the field is there already, how can the right hand rule tell you the correct direction?

Gee this is so confusing, it's one of the reasons why i chose to study maths instead of physics.

So many definitions and weird definitions.

Mentor
2021 Award
the direction of the field is there already
No, the direction of the field is determined by the right-hand rule. The direction of the field is not intrinsic to the physical scenario which consists of the position of the wire and the current through it.

pconstantino
ok but if you havge a wire and a current flowing, it creates the magnetic field around it, how come the right hand rule tells you which direction it is ?? thats the question

right hand rule = human beings
magnetic field = nature

Mentor
2021 Award
ok but if you havge a wire and a current flowing, it creates the magnetic field around it, how come the right hand rule tells you which direction it is ?? thats the question

right hand rule = human beings
magnetic field = nature
Did you understand my example?

How do we determine what direction a magnetic field is? Are there a bunch of little arrows hovering around a current-carying wire to tell us which direction it goes? If not, then how do we determine the direction?

Staff Emeritus
Gold Member

so do you mean that it all comes from the mathematical definition of cross product?

for example, when we take the cross product of 2 vectors via cross product and use determinants, etc... does the right hand rule always match this vector given in the mathematical operation?

Well yeah, I mean if AB = C, then if you start with your fingers in the direction of A, and then curl them in the direction of B, your thumb will point in the direction of C, by the definition of a cross product. If you stick a negative sign in front, you have to use your left hand instead (OR curl your right hand's fingers in the opposite direction, which is an expression of the fact that -(AB) = BA).

if so that seems magical

How so? Given that certain physical quantities can be expressed as vectors, is it really so surprising that certain physical quantities can be expressed as cross products of two other vector quantities? Unless maybe you were speaking in a broader sense that it is magical that we can DO physics at all i.e. that measurable quantities that are useful for describing nature can be related to each other using precise mathematical laws. If so, I would agree that that sometimes seems really amazing, but that is more of a philosophical point, and somehow I don't think it is what you meant. :tongue:

You should REALLY listen to what DaleSpam is saying, but if it helps, here is another example. Physicists CHOOSE to define the direction of the electric field as the direction that a POSITIVE test charge would move if it were placed in that field. However, we could just as easily have chosen to define the direction of the field as the direction that a negative test charge would move if placed in the field. If we did so, we would draw EXACTLY THE SAME field using oppositely-directed vectors (or oppositely directed arrows on the field lines, if we chose to represent the field using field lines). However, nothing about this choice changes anything intrinsic about the field. In particular, it doesn't change the OUTCOME of any experiment conducted in the presence of that field (which is what DaleSpam was trying to illustrate for magnetism). Positive charges would still move one way, the negative charges would still move the opposite way, and they would experience the same forces. EDIT: Just to be absolutely clear and drive this point home: all that would change is the direction we SAY that the field points in.

Similarly, somebody chose to define the "direction" of a magnetic field at a point in space as the direction that a compass needle would point if it were placed at that point in space. (By that, I mean the direction that the end of the needle that always points "north" on Earth would point if placed in that field). They could just as easily have defined it in the opposite sense. Nothing about this change would change any of the measurable results of experiments pertaining to that magnetic field. If we did make this change, we'd probably have to throw a negative sign into the Biot-Savart law or something.

So, directions of vector fields in electromagnetism are not intrinsic to the nature of the fields themselves. They are a matter of CONVENTION. And those conventions are built into the way the mathematical laws pertaining to those fields are formulated.

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Gold Member
ok but if you havge a wire and a current flowing, it creates the magnetic field around it, how come the right hand rule tells you which direction it is ?? thats the question

right hand rule = human beings
magnetic field = nature

You're missing the point. This happens a lot in physics so try to understand what everyone is saying. Nature shows us certain things happen in certain ways. Then, humans think up good ways to REMEMBER how nature works. We don't tell nature "hey, make the magnetic field curl like my fingers like this". We don't tell nature "I like this idea of a cross product, tell God to make magnetic fields obey this mathematical expression" or something similar. We simply analyze the nature of whatever we are studying and see that it always follows a certain form and we try to figure out ways to easily remember it or calculate it.

pconstantino
you are all telling me to listen but I still don't get it, the explanations are too superficial :(

Gold Member
you are all telling me to listen but I still don't get it, the explanations are too superficial :(

They're not meant to be deep! We use the right hand rule only as a device to remember something! If nature was such that magnetic fields or whatever did something else, we'd think up another system to try to remember it! These are purely devices used to remember things. Nature came first, we just thought up ways to remember how nature works. There is nothing profound about it.

It's as if we wanted to invent a clock. Well, we noticed the Sun in the sky. It comes around every so often. In fact, it comes around every 24 hours to the same spot. If we then build a clock that has the ability to count off 24 hour intervals, we've done the exact same thing. We have a device (in this case, a physical device instead of a method of remembering something) that we built because we want to keep track of something nature has shown to be true (that the Sun is periodic). Nothing mysterious. Nothing deep.

Gold Member
What they are trying to say is. Why do you think the magnetic field actually "points" in the direction given by the right hand rule?

It points that way only because we mathematically describe the magnetic field as a vector field that points that way.

By analogy, why do you think that protons have positive charge and electrons have negative charge? Is the proton really made of little symbols that look like plusses? Are electrons made up of little minus signs? No, they have those corresponding charges simply because we define it to be that way.

Things work a certain way in nature, and we find mathematical descriptions for it, but the way we want to define our variables is completely up to us.

pconstantino
But you see, in mathematics does the right hand rule agree with cross product ?

When you take the cross product of two vectors, if you use the right hand rule, will you end up pointing to the same new vector?

Gold Member
But you see, in mathematics does the right hand rule agree with cross product ?

When you take the cross product of two vectors, if you use the right hand rule, will you end up pointing to the same new vector?

Absolutely. That's why we use it, because it works by definition. You will never ever ever find 2 vectors that, when you do the cross product mathematically, will give you an answer that disagrees with the right hand rule (well, unless they're parallel.... which then it doesn't make sense to try to use your hand for anything).

pconstantino
Do you know how this was created?

Gold Member
No but it's fairly intuitive. The cross product mathematically creates something that is perpendicular to two vectors. If you look at your right hand, you'll notice that when put in the correct kinda "right hand rule" positioning, your thumb, fingers, palm, all are pointing in directions that are perpendicular to each other, mimicking the concept behind the cross product.

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Gold Member
But you see, in mathematics does the right hand rule agree with cross product ?

When you take the cross product of two vectors, if you use the right hand rule, will you end up pointing to the same new vector?

Yes, because conventionally we use a right handed co-ordinate system and the cross product is defined this way. Guess which rule we'd be using in a left handed co-ordinate system.

Mentor
2021 Award
The cross product mathematically creates something that is parallel to two vectors.
Pengwuino meant "perpindicular". But elaborating on what he said, the cross product takes 2 vectors and returns a vector which is perpendicular to both.

Pconstantino, let's say that one vector points straight ahead and the other points to the left, then the only vectors which are perpendicular to both are a vector pointing straight up or a vector pointing straight down. How do we pick which one to use? The definition of the cross product is made so that it picks the one determined by the right hand rule.

Btw, you still have not answered the question about how we measure the direction of the magnetic field. So I ask it again here for the third time, and I would request that you not ignore it this time.

Staff Emeritus
Gold Member
But you see, in mathematics does the right hand rule agree with cross product ?

When you take the cross product of two vectors, if you use the right hand rule, will you end up pointing to the same new vector?

I'm frustrated, because I told you this in detail in post #13, as well as elaborating at length upon what other people have been saying here about directions of fields, but it doesn't seem like you read it.

pconstantino
I'm frustrated, because I told you this in detail in post #13, as well as elaborating at length upon what other people have been saying here about directions of fields, but it doesn't seem like you read it.

One of the things I found confusing was, why is the direction of the electric field different from the direction of the force pushing the test charge?

Gold Member
It's only opposite for negative charges. This is because we use the convention of using a positive test charge. Again, this is ALL just definition/convention. There is NOTHING physically important about which way the electric field actually points. There are no invisible little arrows in space telling the electric field, it HAS to point the way we say it points.

Staff Emeritus
Gold Member
pconstantino: Please read dalespam's posts carefully. The reality is that all observables (like the response of a charged particle, or a magnetic dipole) will turn out to agree with the calculation, so long as we are consistent with our choice of hand. Take any situation - like the example dalespam provided - and only use the LH rule instead of the RH rule, and you will end up with the same direction for all observables (note: fields are not observables).

brocks
Do you know how this was created?

I think the point you keep missing is that things like the vector cross product are invented, not discovered. And they are invented specifically to correspond to nature.

Vectors are a fairly new concept in math. They are not a fundamental part of nature, they are just constructs people invented to help describe and predict what happens in nature.

So we can define operations among vectors any way we want.

Start with addition. When you add two vectors, you can just add their components. That seems like a natural way to do things, and it turns out to be useful. So that's how we define vector addition, and everybody's happy.

Now how about vector multiplication? Again, the natural thing might be to just multiply their components, but if you do that, you don't get anything very useful.

But if you multiply the components and then add them together to get a scalar, that does turn out to be useful in describing things like work, so we agree to call that the dot product.

But then somebody else sees that if we multiply the lengths by the sine of the angle, we get another useful quantity. And if we also stipulate a direction perpendicular to the two vectors, we get both the quantity and direction of certain things we find in nature, like torque. So we DEFINE a second way to multiply vectors, and give it a different name, namely the cross product.

It's not that torque or magnetic fields know anything about the cross product; they just do whatever they do. But we notice that they obey some laws we can discover --- we may or may not understand why they obey those laws, but we can observe that they do --- and we INVENT the cross product to help us describe what is happening.

If torque or magnetism followed a different law, then we would have to invent a different operation to describe it. It's not magic that torque follows the cross product, any more than it is magic that a rock falls "down" when you drop it. We just invented "down" to describe the direction we see rocks fall. At a very fundamental level, nobody knows why they fall; they just do. We can observe and measure and predict what will happen with great precision, but we don't really know why gravity does what it does.

So we just invented the cross product to describe what we see happening in nature. And like everybody else is trying to tell you, whether we define the perpendicular direction as right hand or left hand is not important; we would just have to change the sign if we changed hands. What is important is that we pick one, and everybody agrees on it, so we can communicate.

That's really all there is to it.

lalbatros
When I was young, I disliked all these "hands rules".
Therefore, I was used to do without, as far as possible.

For example, if you have a current flowing in a coil,
and if there is a particle moving somehow inside the coil,
then you can easily predict the direction of the force on this particle.
The particle should move in a way that reduces the effect of the current flowing in the coil.
This determine the direction of the force.
If the particle is an electron, it should rotate in the opposite direction compared to the electrons flowing in the coil.
If it worked the other way, the world would be quite unstable.

Unfortunately, I had to cope with all the conventions anyway.
Usually the problems you need to solve for an exam do not tell you the direction of the current in a coil, but instead they state what is the resulting magnetic field. The direction of the magnetic field is conventional and therefore, you need to know all the conventions to be able to solve the problems.
If you want to read scientific papers, you will also need to know these conventions.
In additions, these conventions are useful since -in the end- the they allow you some abstraction: no need to tell how a specific magnetic field was created.

Sorry for my poor english.

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heretolearn
May I be so bold as to direct you to a small study on the Lorentz Force Law. You can find a little info here:
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html" [Broken]
Hope this helps.

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